For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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26
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6answers
5k views

Why should quaternions exist?

Why do quaternions exist? I want to believe they exist, but all I can think of are reasons they should not exist. These are my reasons. The quaternions are defined by the following equation: $$i^...
0
votes
0answers
66 views

Angular Velocity calculation

I am trying to calculate the time derivative of the quaternion from the following paper: Robotics and Biomimetics (ROBIO) See equation 1 below: $\displaystyle{\dot{q}(t)=\frac{1}{2}q(t)\circ\vec{\...
1
vote
1answer
62 views

Calculate hand position from upper-and lower arm's orientation

I've got two unit quaternions in world space representing the lower- and upper arms orientation. The lengths of upper- and lower arm are known. How can i calculate the hand-position relative to the ...
7
votes
0answers
92 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
3
votes
1answer
176 views

Rotation in 4D?

Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is ...
2
votes
1answer
108 views

Expressing unit quaternions in three degrees of freedom

Short version of question: I am trying to use quaternions to avoid gimbal-lock, but I don't know how to express unit quaternions using three degrees of freedom without re-introducing Euler angles and (...
0
votes
0answers
12 views

How to compute angular velocity given a set of unevenly spaced quaternions/direction cosine matrices

I have the time evolution (unevenly spaced) of around 1000 quaternions which provides the transformation from an inertial coordinate system to a body fixed. My goal is to obtain the angular velocity ...
5
votes
1answer
163 views

Quaternion rotation intuition

Say the quaternions real and imaginary part are written as $(q_1, \vec q)$. One useful multiplication property is $qr=(q_1r_1 - \langle\vec q, \vec r\rangle, q_1\vec r + r_1\vec q + \vec q \times \vec ...
2
votes
1answer
88 views

Does symplectic K-theory $KSp$ have products?

The real and unitary topological $K$-theories are cohomology theories defined by the $\Omega$-spectra $KO$ and $K$ respectively. These are multiplicative theories with products deriving from the ...
3
votes
0answers
24 views

Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
2
votes
2answers
132 views

Concise description of why rotation quaternions use half the angle

I'm currently writing the report on my master thesis project, where I use Android sensors to perform inertial navigation in a heavy industrial environment. In my application, I make use of quaternions ...
1
vote
1answer
53 views

Could someone explain the notation of the average of quaternions equation?

The equation has some notation that is difficult to find the meaning for. It is equation (3) in the paper 'Quaternion Averaging' by F. Landis Markley, et al. on page 3 under 'The Average Quaternion'. ...
2
votes
1answer
52 views

What does the subspace of SO(3) corresponding to zero yaw look like

Background : I'm solving an engineering problem where I have to estimate the orientation of a body in 3D space. Usually, I use quaternions to do this, but I have to consider a special case where I ...
4
votes
1answer
141 views

Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by John Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic ...
0
votes
0answers
28 views

Constructing a periodic function around $f(t): R \to R^3$

Definitions: $f(t): R \to R^3$ $\hat{\bigtriangledown}f(t) := \frac{df(t)}{dt} \frac{1}{|\frac{df(t)}{dt}|}$ $\vec{a}e^{\vec{b}x} := \vec{a}\cos(x) + \vec{b}\sin(x)$ $\vec{a}, \vec{b} \in H$ Is ...
0
votes
1answer
71 views

Quaternions and rotation

Basically, I am programming an iOS application where I use attitude of the device in quaternion format. Problem is following: Practically: I have a device that does a measurement #1 of magnetic ...
0
votes
2answers
277 views

Clarification of definition of “inverse” with quaternions

From what I understand, the inverse of a matrix only exists if the matrix is square. I recently learned however that the inverse of a quaternion is the quaternion vector (1xn dimensions) where each ...
1
vote
3answers
52 views

Multiplication of quaternion vectors

Upon watching a lecture on quaternions (Youtube link), I came across the following math: $$(a,\vec{v})(a,- \vec{v})=(a^2+(\vec{v}\cdot \vec{v}),-a\vec{v}+a\vec{v}+(\vec{v}\times \vec{v}))$$ where $a$ ...
2
votes
4answers
109 views

Why is it that with quaternions $ij \neq ji$?

I've been using rotations in 3d space lately for simulations. Today I came across the quaternion, which from what I understand will be a much better alternative to my cross/dot product methods. Now I ...
0
votes
0answers
33 views

quaternions are less versatile than matrix?

I am doing some research looking should I implement quaternions or matrices. What I've seem to come across is that while quaternions can be better for doing smooth rotations and dual quaternions can ...
0
votes
0answers
54 views

4x4 Matrix with homogeneous coordinates

I learn for a linear Algebra exam and I have the task: "What is the $4\times 4$ matrix , a rotation about the $\pi/3$ describes in homogeneous coordinates about the axis? What is the ...
1
vote
1answer
65 views

why is representing rotations through quaternions more compact and quicker than using matrices??

According to the wikipedia page on Quaternions: The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. However, I have to admit,...
2
votes
1answer
75 views

Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$?

Why is it so that a unit quaternion $t$ can be written as $t=\cos(\theta)+u\sin(\theta)$? This question stems from Stillwell's Naive Lie Theory where he states that a quaternion $t$ of absolute value ...
0
votes
1answer
48 views

Equivalance form for Slerp in quaternions interpolation

In all the books I have found that Slerp have two forms: A B I know that all the forms from A are equivalent but I don't know why the forms from A are equivalent with the form from B. Can ...
0
votes
1answer
51 views

Quaternion - trigonometric form - $q=\cos \theta +u \sin \theta$ Components for $u$?

It is proven that a quaternion has the following trigonometric form: $$q=\cos \theta +u \sin \theta.$$ My question is: Which are the components of the $u$? Thanks!
0
votes
2answers
55 views

Quaternion and Euler angles small angle proof

Let's start with a quaternion $q = \begin{bmatrix} q1 & q2 & q3 & q4 \end{bmatrix}^T$. Where $q_4$ is the scalar part, which is equal to: \begin{equation} q_4 = cos(\frac{\alpha}{2}) \end{...
0
votes
1answer
84 views

Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with unit quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = \dot{q}...
2
votes
1answer
110 views

General Linear Group over the quaternions is a topological group

How to show that General Linear Group over the quaternions is a a topological group?
1
vote
1answer
131 views

The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
2
votes
1answer
168 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G Y(t)+...
1
vote
1answer
55 views

Dot product of of quaternion-rotated vectors

I'm reading http://people.csail.mit.edu/bkph/articles/Quaternions.pdf and it says "it is easy to show that the operation preserves dot-products." on the page 3. But how is it done? I tried to make a ...
3
votes
1answer
226 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
0
votes
0answers
211 views

Calculating two rotation angles from xyz coordinates for dummies

This post is a bit verbose so that others who come later may benefit from my thick headedness. I am attempting to construct a primitives composition and constructed solids geometry parser/processor ...
1
vote
1answer
229 views

Multiplication of a quaternion and a scalar to produce a vector?

I am looking at someone else's code, and in it they have a quaternion multiplied with a scalar in order to produce a vector. He used the boost library, and can't find exactly where they defined the ...
0
votes
1answer
41 views

Geometric significance of map, quaternions.

Let $u, v, w \in \mathbb{R}^3$ be a triple of vectors which form an orthonormal basis in $\mathbb{R}^3$ (with the standard orientation). Identify $u, v, w$ with quaternions in the $\mathbb{R}$-linear ...
7
votes
2answers
96 views

Clarify: “$S^0$, $S^1$ and $S^3$ are the only spheres which are also groups”

The zero, one, and three dimensional spheres $S^0$, $S^1$ and $S^3$ are in bijection with the sets $\{a\in \mathbb{K}:|a|=1\}$ for $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$ respectively. The ...
2
votes
0answers
55 views

Factoring quaternion into three parts

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors. What I would like to know is angles (...
2
votes
1answer
47 views

Hasse invariant of quaternions over $\mathbb{Q}_p$

I am trying to compute the Hasse invariant of the quaternion algebra over $\mathbb{Q}_p$. I denote this algebra by $H$, and I'm assuming $p\equiv 3\pmod{4}$. So, $\mathbb{Q}_p(i)$ is an unramified ...
3
votes
1answer
74 views

Why multiplying those 2 quaternions doesn't give the expected result?

Using a left-handed coordinate system, let Q = axisAngle({0,0,1}, 1/4*pi) * axisAngle({0,1,0}, 1/4*pi) be the quaternion representing the rotation "1/8 circle ...
0
votes
1answer
99 views

understanding quaternions - spatial rotations

I would like to know if my understanding about quaternions is correct please: lets say you have a vector in 3d space. You could rotate the x,y and-z frame on a fixed point so that it is parallel with ...
1
vote
1answer
61 views

quaternions - understanding a formula

Quaternions are new for me. I am trying to understand the following formula: What are: $\large{q^x}$ ? I don't think it is a power. $\large{q^t}$ ? just a transposition of the quaternion $q$? ...
0
votes
1answer
59 views

what represents this strange matrix [closed]

But I just encountered a very strange type of matrix. What is the name of this type of matrix? How can it be solved? The final result should be a $4\times 4$ matrix, but I only see $5$ elements on ...
0
votes
0answers
81 views

How do I get the rotation between two rotationmatrices?

I am stuck on a little rotation problem. The problem: I have 2 rotation matrices $A$ and $B$. $A$ and $B$ are relative to the coordinate system O. $A$ and $B$ are Quaternion rotation matrices. I am ...
9
votes
1answer
128 views

Eigenvalues of Group Elements and Quaternions

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the 3-...
2
votes
1answer
38 views

Question about Eigenvalues of group elements

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the 3-...
0
votes
1answer
55 views

What is the mapping from purely imaginary quaternions to a vector in $\mathbb{R}^3$

It is claimed that $q = x{\bf i} + y{\bf j} + z{\bf k}$ has an one to one mapping to a vector $v \in \mathbb{R}^3$ where $v = x \hat i + y \hat j + z \hat k$ But ${\bf i}, {\bf j},{\bf k}$ are ...
3
votes
1answer
127 views

prime divisor propertyfor Hurwitz integers

The Hurwitz integers $\mathcal{H}_{\mathbb{Z}}$ is a particular subset of quaternions. Define: $$ \mathcal{H}_{\mathbb{Z}} = \left\{ a\frac{1+i+j+k}{2}+bi+cj+dk \ | \ a,b,c,d \in \mathbb{Z} \right\} = ...
0
votes
0answers
30 views

Cartesian extremities of a 3d segment

I have a segment in 3d space and I want to calculate its extremities. I know the cartesian coordinates (x,y,z) of the segment's middle point, the segment's length L and the segment's orientation using ...
1
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0answers
42 views

Grothendieck group of a quaternion algebra

Let $\mathcal{O}$ be a maximal order in a quaternion algebra over a number field. Then there is a notion of similarity of (left) $\mathcal{O}$-modules, and similarity classes. The set $S$ of such ...